VaR, CVaR and Time Rules with Elliptical and Asymmetric 1 Stable Distributed Returns Fabio Lamantia2, Sergio Ortobelli3, Svetlozar Rachev4 Abstract This paper proposes several parametric models to compute the portfolio VaR and CVaR in a given temporal horizon and for a given level of confidence. Firstly, we describe extension of the EWMA RiskMetrics model considering conditional elliptically distributed returns. Secondly, we examine several new models based on different stable Paretian distributional hypotheses of return portfolios. Finally, we discuss the applicability of temporal aggregation rules for each VaR and CVaR model proposed. Key words: Elliptical distributions, domain of attraction, stable distribution, time aggregation rules. JEL Classification: G21, C32, C53 1. Introduction This paper presents and discusses risk management models with the same computational complexity of the most used ones in literature. In particular, the paper serves a threefold objective: 1) studying and understanding elliptical EWMA VaR and CVaR models; 2) examining some distributional stable Paretian approaches applied to the evaluation of the risk of a given portfolio; 3) discussing the application and the limits of temporal aggregation rules of EWMA-type VaR and CVaR models. The Value at Risk (VaR) and the Conditional Value at Risk (CVaR) are simple risk measures used by financial institutions to evaluate the market risk exposure of their trading portfolios. The main characteristic of VaR and CVaR is that of synthesizing, in a single value, the possible losses which could occur with a given probability in a given temporal horizon. An important issue in calculating VaR and CVaR is the identification of the so called profit/loss distribution. In the model proposed by RiskMetrics (see Longerstaey and Zangari, 1996), the main assumption is that the profit/loss distribution, conditional upon the portfolio standard deviation, is Gaussian. The main consequence of this hypothesis is that the percentiles and the conditional expected loss, therefore VaR and CVaR, can be simply calculated by multiplying the portfolio standard deviation times a constant which is function of the given confidence level. On the other hand, the possibility of on-line “Gaussian” VaR and CVaR computation have represented the main “success” of these parametric models. As a matter of fact, in the last years there has been a growth in the number of those investors who 1 We wish to express our most sincere gratitude to Gennady Samorodnitsky for his useful advices and suggestions that help us to overcome the limits of a preliminary version of the paper. We also are grateful to Hercules Vladimirou and to several anonymous readers for helpful comments. 2 Researcher, Department SA, University of Calabria, Italy. 3 Researcher, Department MSIA, University of Bergamo, Italy. Sergio Ortobelli’s research has been partially supported under Murst 40%-60% 2004, 2005, 2006 and CNR-MIUR-Legge 95/95. 4 Contact author: e-mail: [email protected] Chair of Econometrics, Statistics and Mathematical Finance, School of Economics and Business Engineering, University of Karlsruhe Kollegium am Schloss, Bau II, 20.12, R210 Postfach 6980, D-76128, Karlsruhe (Germany). Svetlozar Rachev is also Professor Applied Probability University of California at Santa Barbara. Svetlozar Rachev's research has been supported by grants from Division of Mathematical, Life and Physical Sciences, College of Letters and Science, University of California, Santa Barbara and the Deutsche Forschungsgemeinschaft. 1 prefer manage on-line their own portfolios. Moreover, practitioners use to forecast the weekly, monthly, yearly losses under risk by scaling with opportune factor daily “Gaussian” VaR and CVaR estimates. However, although this temporal rule is very useful by a practical point of view, it is not generally valid except when we consider independent Gaussian distributed returns. In addition, many empirical studies show that the return conditional distributions diverge from the Gaussian one. In particular, it has been observed that the profit/loss distributions present asymmetries and fat tails. As shown in Longerstaey and Zangari (1996), the VaR calculated under the normal assumption underestimates the actual risk, given that the distribution of the observed financial series are leptokurtic with respect to those implied by a conditional normal distribution. This paper presents several alternative models for the calculation of VaR and CVaR taking into consideration the skewness and the kurtosis (fat tail effect) that mark the empirical profit/loss distributions. In order to maintain the simplicity of the RiskMetrics model we first extend it to an exponential weighted moving average (EWMA) model with conditional elliptically distributed returns and finite variance. In particular, we discuss the opportunity of using temporal rules of aggregated EWMA models that have not been correctly justified by RiskMetrics researchers (see Longerstaey and Zangari, 1996). Thus, we show that time aggregation rules can be used only when we assume independently distributed returns to approximate the future VaR and CVAR estimates, and then we focus our attention on returns with a conditional multivariate elliptical distributions. Secondly, we propose many stable Paretian VaR and CVaR models. Several empirical and theoretical studies on the asymptotic behavior of financial returns (see, among others, Mandelbrot, 1963a-b, Fama, 1965) justify the assumption of stable paretian distributed returns. Therefore, many stable models have been proposed in recent literature to study financial applications of the stable distributions (see Samorodnitsky and Taqqu, 1994, Rachev and Mittnik, 2000, and references therein). In particular, Stoyanov et al., 2006 have proved some closed form solutions to compute conditional value at risk of a given elliptical and/or stable paretian distribution. However, all stable VaR models recently proposed in financial literature either describe simulating models (see Rachev et al, 2003) or propose models that do not take into account the dependence structure among asset returns and their autoregressive behavior (see, for example, Mittnik, et al 2002). In contrast, in this paper we first present two parametric autoregressive stable models and we propose two relative time aggregation rules for the associated unconditional models where the return series are independent and identically distributed (i.i.d.). In the first stable model we consider conditional α-stable sub-Gaussian distributed returns. The joint stable sub-Gaussian family is an elliptical family that has been recently used in portfolio theory (see Rachev, et al, 2004, Mittnik, et al, 2002). As for the elliptical unconditional model with finite variance, we describe particular temporal rules of VaR and CVaR. In order to consider the asymmetry of financial series, we assume conditional jointly asymmetric α-stable distributed returns. The asymmetric stable model results from a new conditional version of the stable three fund separation model proposed by Rachev, et al, 2004 to study the portfolio choice problem with asymmetric returns. In this case too, when we assume that the returns series are i.i.d., we obtain further time aggregation rules for VaR and CVaR. The paper is organized as follows: in section 2 we formalize the RiskMetrics model and its elliptical extension. Section 3 introduces alternative approaches to the EWMA VaR and CVaR models with finite variance. Finally, we briefly summarize the paper. 2. RiskMetrics approach and elliptical EWMA models with finite variance Value at Risk is the maximum loss among the best θ % cases that could occur in a given 2 − temporal horizon. If we denote with τ the investor's temporal horizon, with WWtt+τ the profit/loss realized in the interval [t,t+τ] and with θ the level of confidence, then VaR is given by the loss such that, −= −≤>−θ VaRθτ,[tttt ,++ ] ()inf|Pr1 W τ W{ q() W tt + τ W q } . (1) Hence, the VaR is the percentile at the (1-θ)% of the profit/loss distribution in the interval [t,t+τ]. The temporal horizon τ and the level of confidence θ are chosen by the investor. The choice of τ depends on the frequency with which the investor wishes to control his/her investment. Alternatively to VaR, the recent literature on risk measures (see Szegö, 2004) have proposed the conditional value at risk (CVaR), also called expected shortfall or expected tail loss, to evaluate the exposure to market risks. The conditional value at risk is a coherent risk measure i.e. it is a positively homogenous, translation invariant, subadditive and monotone risk measure. Even if there is no doubt that VaRϑ provides useful information, VaRϑ is not a coherent risk measure (see Artzner et al., 1999 and Stoyanov et al., 2006) and it cannot offer exhaustive information about the expected future losses5. The conditional value at risk measures the expected value of profit/loss given that the Value at Risk has not been exceeded, that is 1 1−θ CVaR() W−= W VaR() W − W dq (2) θτ,[tt ,++ ] t τ t−θ ∫ q ,[ tt , + τ ] t + τ t 1 0 and if we assume a continuous distribution for the profit/loss distribution, we obtain −= − −≤ CVaRθτ,[tttt ,++ ]() W τ W E() W tttt + τ W | W + τ W VaR θτ , t +. The RiskMetrics model assumes that τ = the conditional distribution of the continuously compounded return Rttt() log()WW+τ / is a Gaussian law. In particular, RiskMetrics simplifies the VaR calculation for those portfolios = with many assets. If we denote with ww[1 ,..., wn ]' the vector of the positions taken in n assets forming the portfolio, then the return portfolio at time t+1 is given by n = zwz(),1pt++∑ iit ,1, i=1 = where zit,1++log()PP it ,1 / it , is the (continuously compounded) return of i-th asset during the period [t,t+1], and Pit, is the price of i-th asset at time t. RiskMetrics assumes that within a short period of time, the expected return is null and that the return vector = zzttnt+++1 1,1,..., z ,1 ' follows a conditional joint Gaussian distribution.